As mentioned already, their ages are different. Cyborg confirmed Changeling's age in issue 4 by saying, "he's only 16" during a drinking scene. The audience can assume his age from his maturity, but there are other small details here and there. For one, the other members treat him like he's a kid. Another thing is other Titans mention his age when Terra comes around. There's a scene when Terra and Beast Boy are home-schooled as well, indicating again that he's probably 16. If he's the same age as Terra, then it would also confirm that he's 16. In the tv series, it's theorized that's he's 14-15. Just like the comics, he's supposed to be the youngest member, so it only makes sense for the age range to be 15 and younger. In the most recent adaptation (a live-action version), Beast Boy is 17 with Raven being younger than him by two years.
The main difference between the tv comic version and the tv series version is that they have different names. In the comics, Beast Boy goes by Changeling while in the tv series, he's Beast Boy. His real name, Garfield Logan, is used a lot in the comics, but in the tv series, the Teen Titans didn't find out his real name until they met the Doom Patrol in later seasons. Comic Beast Boy, aka Changeling, does end up going by Beast Boy later in the series. Other than their name, I don't share any similarities among their character design. In the tv series, his jumpsuit is black and purple. In the comics, his jumpsuit is red and white. His hairstyle is even different; it's short and spiky in the tv series, while boy-next-door prim and proper in the comics. TV creators portrayed TV Beast Boy as a young teenager, so it makes sense he has no abs, but Comic Beast Boy has a strong physique that comes from heavy gym time.
The only Titan with a copy and paste personality between the comics and the tv series is Beast Boy. Honestly, there's no difference between his character's personality, even down to the loneliness and inadequateness that they both feel. Beast Boy is playful, energetic, and often immature. While both versions struggle to be serious, Comic Beast Boy struggles with it more to the point where he might refuse to be serious. TV Beast Boy has his serious moments and even has lashed at people when they haven't taken him seriously. I think the only difference between the way they handle Beast Boy's sadness. Comic Beast Boy's sadness is more outwardly, making appearances a lot when he's alone or when something affects him. He tends to take his own emotions above other members whenever he becomes sad. The tv series developed TV Beast Boy's sadness similarly, but it avoids taking a serious stance like Depression to relate more to the younger audience without a scandal. Despite the trauma that they both go through, they tend to stay optimistic, friendly, and upbeat. They're the mood maker of the group and go out of their way to try to make their members happy.
There are a couple of differences in Beast Boy's backstory from the comics and the tv series. The origin story of his powers is the same in the comics and the tv series. While traveling with his parents, he contacts an illness called Sakutia. To heal him, they give him a serum made from a green monkey. Similarly, his birth parents are dead in both versions. It's what happens to Beast Boy afterward that the two stories start to diverge. In the comics, Elastic Girl and Mento, members of Doom Patrol, adopt him. They got married, but soon after, Elastic Girl died, leaving Beast Boy in the hands of Mento. Beast Boy holds resentment against Mento because he goes missing for a long time and doesn't seem to care about Beast Boy after Elastic Girl died. In the tv series, no one adopted him, but instead, he sought out Doom Patrol himself to join them. He ended up leaving Doom Patrol because of the harsh discipline from Mento. After leaving Doom Patrol, he runs into the others, and they form Teen Titans. While in TV Beast Boy's case he was alone for most of his life, Comic Beast Boy had it rough because of all the death he had to deal with at a young age. Some more differences are the Doom Patrol. Elastic Girl and Mento weren't married in the tv series, and Mento was the leader. In the comics, Mento wasn't the leader.
Robin: In my opinion, Robin is the member Beast Boy's the least close to, which isn't that concerning even though Robin is the leader. In the comics, he put Robin on a pedestal, being blinded by his achievements that he didn't acknowledge that Robin had flaws like everyone else. Whenever he was self-deprecating, he would compare himself to Robin. On the other hand, Robin wanted more from Beast Boy, so he tended to push him with force, hoping to make him improve. This attitude caused stress in their relationship because Robin became an authority figure to Beast Boy. Even though they had problems, Beast Boy was upset when Robin started distancing himself. In the tv series, they aren't that close. They don't hang out together, and they don't share any hobbies. They also tend to have problems because they have different personalities. The respect and trust each other better than the comics, with both being proud of the other.
Starfire: They have an almost non-existent relationship in the comics, lacking a presence to the point that I don't remember them interacting besides battles. He's a naturally flirty character, so that might be some of their interactions. Other than that, there's nothing much to talk about concerning their relationship. In the tv series, they have an underrated relationship. As two of my favorite characters, they are close and relate to each other a lot. They have similar personalities, and both go through a type of sadness that affects them when they least expect it. Starfire's one of the only people to be verbal about Beast Boy going missing and can tell whenever he's down. Beast Boy enjoys having someone who can match his enthusiasm and brings Starfire along the ride of some of his crazy ideas. They also share in the fact that both of them are affected by their past a lot. Since they have similar pain, they tend to confide in each other and comfort the other.
Raven: In the comics, I don't remember them interacting outside of battle. They both lived in the Titans' Tower and probably interacted daily, but the comics didn't do any development on their relationship. Their relationship is strictly platonic in The New Teen Titans. In later comic series, they develop feelings for each other. They get together shortly afterward. They have a couple of bumps along the way. For instance, his past relationship with Terra and her fear of her powers. While they do end up breaking up, they can't stay away and decide to get back together despite Raven's fears. I will provide a link of where I read the development of their relationship; while it comes from an online forum like Quora, I do feel it has some truth to it as I looked up some of the comics mentioned. In the tv series, similar to the comics, their relationship stays platonic. There's not much indicating in the tv series that they planned to make them ever romantic, but a series created by Cartoon Network called Teen Titans' GO! has provided a lot of content with a romantic subtext. Regardless if they were ever supposed to be romantic or not, they're close. This is where opposites attract plays a part in their dynamic. They are very different, but Beast Boy is determined to get closer to Raven and probably hasn't realized that he already did. He goes out of his way to try to cheer her up, and he's constantly trying to get her to hang out with him. Beast Boy has shown time and time again that he gets affected whenever she's in trouble, for instance, when he turns into a beast to protect her, and that he's not afraid to go the extra distance for her. He also gets thrilled whenever she reacts to him even sometimes when it's a negative reaction.
Cyborg: In the comics, they're best friends. Cyborg is probably the only member that Beast Boy is close to in the comics, and they hung out. They both live in the Titans' Tower and had serious conversations whenever one of them was upset. Since their personalities were a bit different in the comics, they didn't automatically get along, but with persistence, Beast Boy got Cyborg to open up to him. They were close enough that Beast Boy was rarely on the opposite side of Cyborg's anger, and he was one of the few people who could call Cyborg out on his actions. In the tv series, they are the dynamic duo. It's rare to see them separate besides their episodes; they do everything together. While they are certain activities (like cars for Cyborg) that they don't together, you can still see them hanging out or questioning the other about what they're doing. Beast Boy respects Cyborg, but he's close to Cyborg because Cyborg doesn't look down at him or write him off like some kid. They're on the same level, and Cyborg even trusts him with larger things or being able to do his battles. From what I've seen, Cyborg and Beast Boy tend to have a very close bond in most versions. I do think he's closest to Cyborg.
Terra: TBD
Wonder Girl/Kid Flash: Wonder Girl tends to take an older sister role with Beast Boy, nagging him for his actions, being there when he might need a hug, etc. Their relationship doesn't have much development, but she reacts fairly well to his jokes. The only noteworthy thing about his relationship with Kid Flash is the fact that Beast Boy aggravates him. Their only interactions usually end with Kid Flash wanting to punch Beast Boy in the face. Even though Kid Flash does appear in the tv series, he doesn't interact with Beast Boy.